Tag for questions about set-valued functions and their properties.
Questions tagged [set-valued-analysis]
76 questions
6
votes
0 answers
Monotone (measurable) selection theorem
Given a Polish space $X$ and a set valued map $\Psi:X\to 2^X$ we way $\Psi$ is weakly Borel if for every open $U\subset X$, $$
\Psi^{-1}(U):=\{x\,|\,\Psi(x)\cap U\ne\varnothing\}\in\mathcal{B}(X).
$$
With the additional assumption that $\Psi(x)$ is…
APP
- 439
- 2
- 8
6
votes
1 answer
Measurable argmax correspondence on probability spaces
I'm trying to wrap my head around the "Measurable maximum theorem", Thm 14.91 in "A hitchhiker's guide to infinite dimensional analysis" by Aliprantis & Border.
I wonder if I can use it in a case when the underlying measurable space is a probability…
user202542
- 931
5
votes
1 answer
Prove the (path-) connectedness of the graph of a compact- and convex-valued upper hemi-continuous correspondence.
Let $\Gamma: [0, 1] \to \mathbb{R}$ be a compact- and convex-valued, upper hemi-continuous correspondence. Prove that the graph of $\Gamma$ is a connected set. Is it path-connected?
This is what I have so far:
Proof $\space$ We first show that…
Beerus
- 2,939
5
votes
0 answers
algebraic or homotopical proof for Kakutani fixed point theorem
As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy theory or if it has at least some applications in…
Rahman
- 1,179
4
votes
1 answer
A question on the directional derivative in a real Banach space
Let $K$ be a convex and compact set in the real Banach space $(\mathbb{R}^n, |\cdot|_{\infty})$. Here, the max-norm is defined as $|v|_{\infty}=\max_{i}|v_{i}|$ and the distance function $d_{K}(\cdot)$ is defined to be $d_{K}(x):=\inf_{y \in…
HTChoi
- 105
4
votes
1 answer
Subdifferential of a proper convex function: is it upper-hemicontinuous?
I am fairly new with the concept of upper-hemicontinuity, i.e.
Let $X \subseteq E^{n}$, $Y \subseteq E^{m}$ and $\Psi: X \rightrightarrows Y$ be a set-valued map. $\Psi$ is upper
hemicontinuous at $x_{0} \in X$ if, for every open set $V…
Julien V
- 165
3
votes
0 answers
Question about the equivalence of three versions of Closed Graph Theorem
I am studying set-valued analysis, and I saw three version of this Closed Graph Theorem.
Version 1: (what I was taught in class)
Let $\Gamma: X \to Y$ be a correspondence. If $\Gamma$ is closed-valued and upper hemi-continuous, then $Gr(\Gamma)$ is…
Beerus
- 2,939
3
votes
0 answers
Continuity of a set-valued function
Let $f:\mathbb{R}^{n}\rightrightarrows \mathbb{R}^{m}$ be a set-valued
function defined by
\begin{equation*}
f\left( x\right) =\left\{ y\in \mathbb{R}^{m}:g\left( x\right) +h\left(
x\right) ^{T}y\leq 0\right\} \text{,}
\end{equation*}
where…
UnclePetros
- 63
3
votes
0 answers
On the continuity of the funciton $w \mapsto \mu(\{x \in X \mid w^\top v(x) > t\})$
Let $X$ be a measurable space and let $\mu$ be a probability measure on $X$ (assumed to be non-atomic, in case that helps). Let $v:X \to \mathbb R^k$ be a $\mu$-integrable function and for every $t \in [0,b]$ (with $b>0$ fixed), consider the…
dohmatob
- 9,753
3
votes
0 answers
Differntial inclusions that are single valued near hyperbolic rest point: Hartman Grobman still fine?
Suppose for $x\in \mathbb{R}^m$ we have a well behaved differential inclusion $\dot x \in F(x)$, where $F(x)$ is a Marchaud map, i.e. u.s.c. with compact convex values and linear growth condition: there exists $c>0$ s.t. $$\sup\{\|y\|:\, y\in F(x)…
Clempe
- 61
3
votes
0 answers
References about the historical move from set-valued maps to single-valued maps, and from curves to functions.
In
Aubin, Jean-Pierre, and Hélène Frankowska. Set-valued analysis. Springer Science & Business Media, 2009.
We read that historically, set-valued analysis was being developed by Kuratowski and Hausdorff, along with other Polish and French…
jadn
- 459
- 2
- 10
3
votes
2 answers
Does the closure of a strict sublevel set of a continuous function equal a non-strict sublevel set?
Let $X \subset \mathbb{R}^{n}$ be closed and $f$ be a continuous real valued function on $X$.
Consider now the sets
\begin{align}
S_{<} = \{ x \in X : f(x) < 0 \}, &&
S_{\le} = \{ x \in X : f(x) \le 0 \}.
\end{align}
Is it generally true that if $…
node
- 203
3
votes
2 answers
Passing to weak-strong limit in pointwise inclusions
Let $F:\mathbb R^m\rightrightarrows\mathbb R^n$ be a set-valued map (or multi-function, correspondence) with $F(x)\ne\emptyset$ for all $x\in \mathbb R^m$.
Let $I\subset\mathbb R$ be an interval.
Let be sequences of functions $(y_n)$ and $(x_n)$…
daw
- 54,637
- 2
- 44
- 85
3
votes
1 answer
Graph of the composition of a continuous function and a correspondence with a graph homeomorphic to its domain
Let $\phi: \mathbb{R}^m \leadsto \mathbb{R}^n $ be an upper hemi-continuous correspondence, $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous function. If the graph of $\phi$, $\{(x,y) \in \mathbb{R}^{m+n} \mid y \in \phi(x) \}$, is homeomorphic to…
Metta World Peace
- 6,683
3
votes
0 answers
Is an upper semicontinuous correspondence weakly measurable?
Definition 1
Let $F:X\to2^Y$ be a set-valued map from a metric space to the subsets of another metric space. We say it is upper semi-continuous (USC) if for every $\epsilon$ and every $x_0\in X$ there exists $\delta$ such that…
MickG
- 9,085